To determine where the average rate of change of the function \( f(x) \) is greatest, we need to examine the intervals over which the changes in the function's values occur, particularly focusing on the slopes between the endpoints of each interval.
- The average rate of change over an interval \([a, b]\) is given by the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
If you were to analyze each of the given intervals based on the graph (assuming the graph shows measurable points for \( f(x) \)), you would calculate the average rate of change for each one:
- For the interval \([0, 1]\)
- For the interval \([1, 4]\)
- For the interval \([4, 5]\)
- For the interval \([5, 6]\)
Without the actual image of the graph, I can't determine the exact outputs. Generally, you would compare the calculated average rates across these intervals:
- If the slope between endpoints is steep, that indicates a greater rate of change.
- Look for the interval where the end values of \( f(b) \) and \( f(a) \) show a significant vertical change when compared to the horizontal distance.
If any of the intervals, such as \([1,4]\) or \([5,6]\), show a notably steeper change as compared to others (like possibly having a large difference in \( f(x) \) values), you would choose that interval as having the greatest average rate of change.
Please refer back to the graph to observe which of these intervals truly displays the highest steepness in the corresponding segments for a final answer.
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